Moment generating functions

• What Is the MGF of the Ratio X/Y?
  There is no universal MGF for X/Y: it depends on the joint distribution, and it may fail to exist near zero even when X and Y have MGFs.
• What Is the Moment-Generating Function of an Exponential Distribution?
  Derive the exponential distribution’s moment-generating function, identify where it exists, and use it to calculate moments and sums of independent variables.
• What Is the Moment-Generating Function of a Beta Distribution?
  For X ~ Beta(alpha, beta), the moment-generating function is 1F1(alpha; alpha + beta; t). Derive it from the beta integral and use it to find moments.
• What Is the MGF of X − Y?
  For independent random variables, the MGF of X−Y is M_X(t)M_Y(−t); see how dependence and existence domains change the result.
• What Is the MGF of a Double Exponential Random Variable?
  For a Laplace (double exponential) random variable, the MGF is e^(μt)/(1 − b²t²), valid only for |t| < 1/b.
• Is the Sample Mean of a Poisson Distribution Normal?
  For iid Poisson observations, the sample mean is a scaled Poisson variable, not exactly normal; the CLT gives its large-sample normal approximation.
• How to Compute the MGF of a Compound Poisson Distribution
  Compute a compound Poisson MGF by conditioning on the Poisson count: the result is exp(lambda times (the jump MGF minus one)).